Numerical range and numerical radius

Let $H$ be a Hilbert space equipped with the inner product (x,y), and let $B(H)$ be the algebra of bounded linear operators acting on $H$. The numerical range (also known as the field of values) $W(A)$ of $A in B(H)$ is the collection of all complex numbers of the form $(Ax, x)$, where $x$ is a unit vector in $H$, and the numerical radius $r(A)$ of A is the radius of the smallest circle centered at the origin containing $W(A).$ The study of numerical range and numerical radius has an extensive history, and there is a great deal of current research on these concepts and their generalizations. In particular, the subject has connections and applications to various areas including $C^*-$ algebras, iteration methods, several operator theory, dilation theory, Krein space operators, factorizations of matrix polynomials, unitary similarity, etc. (e.g., see [AnL], [Ar], [Ax], [Ba], [BiFL], [BoD1], [BoD2], [F], [Ha1], [Ha2], [HJ, Chapter 1], [Ist, Chapter 6], [LR], [LTU], [Mo], [P], [S], and their reference.) Further research on this topic may lead the discovery of more connections and applications of the theory of numerical ranges to other subjects.

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